Triangles inscribed inside a decagon.

Total number of triangles that can be drawn by joining any three of the 10 vertices of the decagon $\ = \begin{pmatrix} 10 \\ 3 \end{pmatrix} = 120$ .

Number of triangles that share one common side $\ = \$ Number of ways in which two adjacent points can be selected $\times$ Number of ways in which another point can be selected which is non adjacent to either of them $\ = 10 \cdot \begin{pmatrix} 6 \\ 1 \end{pmatrix} = 60$

Number of triangles that share two common sides $\ = 10$ .

So, required total $\ = 120 - (60+10) = \boxed{50}$ .